Heritage High School

Algebra 1 – Mr. Messier

Week 7: 5/18-5/25

Included in this packet: Solving Quadratics by Using the Quadratic Formula Notes Template Solving Quadratics by Using the Quadratic Formula Notes Key Determining the Number of Solutions Using the Discriminant Notes Template Determining the Number of Solutions Using the Discriminant Notes Key Solve Using Any Method Summary Notes Key 9.5 Textbook Pages You should treat these notes as if you were copying them down from the board – use the key to copy the notes onto your template (or if you don’t have a printer, you can just copy onto any paper that you have). While going through each problem, ask yourself if it makes sense or not, if you have questions, please view the lesson videos on my site, the tutorial videos in the Dynamic ebook on Clever or reach out to me!

Assignments to be submitted by 9:00 am on Tuesday, May 26:

Read p. 516-520 (hint: use the Dynamic e-book on Clever to see the video tutorials)

Complete Big Ideas – 9.5 pg. 521-524 #10, 11, 14, 15, 18, 20, 37, 39, 40, 42-46, 74

Little to No Technology Access- You may take a pic/scan your assignment and email it to your Algebra 1 teacher – messierd@luhsd.net (this is preferred) or Packets will be accepted between 12-3pm on Monday at the main office. If you cannot do this then you need to call the main admin office to make other arrangements. Access to Technology- The preferred method to complete your homework is electronically through Clever.

Solving Quadratic Equations using the Quadratic Formula Name: _________________________________ Date: _________________________________

You can find the solutions to any quadratic equation in standard form (𝑎𝑥 + 𝑏𝑥 + 𝑐) using

The Quadratic Formula.

2 4

2

b b acx

a

Ex 1: Solve 22 5 3 0x x using the quadratic formula.

Identify a, b, and c.

Substitute those values into the Quadratic Formula

Simplify inside the radical. Be sure to follow order of operations!

Evaluate the square root.

Reduce the fractions if possible. Identify both solutions.

Ex 2: 2 6 5 0x x Ex 3: 212 4 5 0x x

Ex 4: 23 2 7 0x x Ex 5: 24 4 1x x

You Try!

1: 2 8 12x x 2: 23 10 9x x

3: 23 6 9 0x x 4: 22 18 0x

Determining the number of solutions using the discriminant Name: _________________________________ Date: _________________________________

The expression under the radical in the Quadratic Formula, is called the discriminant.

2 4

2

b b acx

a

Interpreting the discriminant

2 4 0b ac

Two real solutions Two x-intercepts

2 4 0b ac

One real solution One x-intercept

2 4 0b ac

No real solutions No x-intercepts

Determine the number of real solutions for each quadratic equation.

Ex 1: 2 8 3 0x x

Identify a, b, and c.

Substitute those values into the discriminant of the Quadratic Equation

Simplify. Be sure to follow order of operations!

Determine the number of solutions (x-intercepts). 2 4 0b ac Two solutions (two x-intercepts) 2 4 0b ac One solution (one x-intercept) 2 4 0b ac No solutions (no x-intercepts)

The Discriminant

Ex 2: 29 1 6x x

You Try!

3: 26 2 1x x 4: 2 4 4 0x x

Use the discriminant to find the number of x-intercepts.

Ex 1: 2 14 2y x x Ex 2: 26 2 1f x x x

You Try!

3: 2 6y x x 4: 2f x x x

Section 9.5 Solving Quadratic Equations Using the Quadratic Formula 521

Exercises9.5 Dynamic Solutions available at BigIdeasMath.com

In Exercises 3–8, write the equation in standard form. Then identify the values of a, b, and c that you would use to solve the equation using the Quadratic Formula.

3. x2 = 7x 4. x2 − 4x = −12

5. −2x2 + 1 = 5x 6. 3x + 2 = 4x2

7. 4 − 3x = −x2 + 3x 8. −8x − 1 = 3x2 + 2

In Exercises 9–22, solve the equation using the Quadratic Formula. Round your solutions to the nearest tenth, if necessary. (See Example 1.)

9. x2 − 12x + 36 = 0 10. x2 + 7x + 16 = 0

11. x2 − 10x − 11 = 0 12. 2x2 − x − 1 = 0

13. 2x2 − 6x + 5 = 0 14. 9x2 − 6x + 1 = 0

15. 6x2 − 13x = −6 16. −3x2 + 6x = 4

17. 1 − 8x = −16x2 18. x2 − 5x + 3 = 0

19. x2 + 2x = 9 20. 5x2 − 2 = 4x

21. 2x2 + 9x + 7 = 3 22. 8x2 + 8 = 6 − 9x

23. MODELING WITH MATHEMATICS A dolphin

jumps out of the water, as shown in the diagram.

The function h = −16t2 + 26t models the height h (in feet) of the dolphin after t seconds. After how

many seconds is the dolphin at a height of 5 feet? (See Example 2.)

24. MODELING WITH MATHEMATICS The amount of trout

y (in tons) caught in a lake from 1995 to 2014 can be

modeled by the equation y = −0.08x2 + 1.6x + 10,

where x is the number of years since 1995.

a. When were about 15 tons of trout caught in

the lake?

b. Do you think this model can be used to determine

the amounts of trout caught in future years?

Explain your reasoning.

In Exercises 25–30, determine the number of real solutions of the equation. (See Example 3.)

25. x2 − 6x + 10 = 0 26. x2 − 5x − 3 = 0

27. 2x2 − 12x = −18 28. 4x2 = 4x − 1

29. − 1 — 4 x2 + 4x = −2 30. −5x2 + 8x = 9

In Exercises 31–36, fi nd the number of x-intercepts of the graph of the function. (See Example 4.)

31. y = x2 + 5x − 1

32. y = 4x2 + 4x + 1

33. y = −6x2 + 3x − 4

34. y = −x2 + 5x + 13

35. f (x) = 4x2 + 3x − 6

36. f (x) = 2x2 + 8x + 8

In Exercises 37–44, solve the equation using any method. Explain your choice of method. (See Example 5.)

37. −10x2 + 13x = 4 38. x2 − 3x − 40 = 0

39. x2 + 6x = 5 40. −5x2 = −25

41. x2 + x − 12 = 0 42. x2 − 4x + 1 = 0

43. 4x2 − x = 17 44. x2 + 6x + 9 = 16

Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics

1. VOCABULARY What formula can you use to solve any quadratic equation? Write the formula.

2. VOCABULARY In the Quadratic Formula, what is the discriminant? What does the value of the

discriminant determine?

Vocabulary and Core Concept CheckVocabulary and Core Concept Check

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522 Chapter 9 Solving Quadratic Equations

45. ERROR ANALYSIS Describe and correct the error

in solving the equation 3x2 − 7x − 6 = 0 using the

Quadratic Formula.

x = −7 ± √——

(−7)2 − 4(3)(−6) ——— 2(3)

= −7 ± √—

121 —— 6

x = 2 — 3

and x = −3

✗

46. ERROR ANALYSIS Describe and correct the error

in solving the equation −2x2 + 9x = 4 using the

Quadratic Formula.

x = −9 ± √——

92 − 4(−2)(4) —— 2(−2)

= −9 ± √—

113 —— −4

x ≈ −0.41 and x ≈ 4.91

✗

47. MODELING WITH MATHEMATICS A fountain shoots

a water arc that can be modeled by the graph of the

equation y = −0.006x2 + 1.2x + 10, where x is the

horizontal distance (in feet) from the river’s north

shore and y is the height (in feet) above the river. Does

the water arc reach a height of 50 feet? If so, about

how far from the north shore is the water arc 50 feet

above the water?

20

20 x

y

48. MODELING WITH MATHEMATICS Between the

months of April and September, the number y of hours

of daylight per day in Seattle, Washington, can be

modeled by y = −0.00046x2 + 0.076x + 13, where x

is the number of days since April 1.

a. Do any of the days between April and September in

Seattle have 17 hours of daylight? If so, how many?

b. Do any of the days between April and September in

Seattle have 14 hours of daylight? If so, how many?

49. MAKING AN ARGUMENT Your friend uses the

discriminant of the equation 2x2 − 5x − 2 = −11 and

determines that the equation has two real solutions. Is

your friend correct? Explain your reasoning.

50. MODELING WITH MATHEMATICS The frame of the

tent shown is defi ned by a rectangular base and two

parabolic arches that connect the opposite corners of

the base. The graph of y = −0.18x2 + 1.6x models

the height y (in feet) of one of the arches x feet

along the diagonal of

the base. Can a child

who is 4 feet tall walk

under one of the arches

without having to bend

over? Explain.

MATHEMATICAL CONNECTIONS In Exercises 51 and 52, use the given area A of the rectangle to fi nd the value of x. Then give the dimensions of the rectangle.

51. A = 91 m2

(x + 2) m

(2x + 3) m

52. A = 209 ft2

(4x − 5) ft

(4x + 3) ft

COMPARING METHODS In Exercises 53 and 54, solve the equation by (a) graphing, (b) factoring, and (c) using the Quadratic Formula. Which method do you prefer? Explain your reasoning.

53. x2 + 4x + 4 = 0 54. 3x2 + 11x + 6 = 0

55. REASONING How many solutions does the equation

ax2 + bx + c = 0 have when a and c have different

signs? Explain your reasoning.

56. REASONING When the discriminant is a perfect

square, are the solutions of ax2 + bx + c = 0 rational

or irrational? (Assume a, b, and c are integers.)

Explain your reasoning.

REASONING In Exercises 57–59, give a value of c for which the equation has (a) two solutions, (b) one solution, and (c) no solutions.

57. x2 − 2x + c = 0

58. x2 − 8x + c = 0

59. 4x2 + 12x + c = 0

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524 Chapter 9 Solving Quadratic Equations

74. HOW DO YOU SEE IT? Match each graph with its

discriminant. Explain your reasoning.

A.

x

y

B.

x

y

C.

x

y

a. b2 − 4ac > 0

b. b2 − 4ac = 0

c. b2 − 4ac < 0

75. CRITICAL THINKING You are trying to hang a tire

swing. To get the rope over a tree branch that is

15 feet high, you tie the rope to a weight and throw

it over the branch. You release the weight at a height

s0 of 5.5 feet. What is the minimum initial vertical

velocity v0 needed to reach the branch? (Hint: Use

the equation h = −16t2 + v0t + s0.)

76. THOUGHT PROVOKING Consider the graph of the

standard form of a quadratic function y = ax2 + bx + c. Then consider the Quadratic Formula as given by

x = − b —

2a ±

√—

b2 − 4ac —

2a .

Write a graphical interpretation of the two parts of

this formula.

77. ANALYZING RELATIONSHIPS Find the sum and

product of −b + √

— b2 − 4ac ——

2a and

−b − √—

b2 − 4ac ——

2a .

Then write a quadratic equation whose solutions have

a sum of 2 and a product of 1 —

2 .

78. WRITING A FORMULA Derive a formula that can

be used to fi nd solutions of equations that have the

form ax2 + x + c = 0. Use your formula to solve

−2x2 + x + 8 = 0.

79. MULTIPLE REPRESENTATIONS If p is a solution of a

quadratic equation ax2 + bx + c = 0, then (x − p) is

a factor of ax2 + bx + c.

a. Copy and complete the table for each pair of

solutions.

Solutions Factors Quadratic equation

3, 4 (x − 3), (x − 4) x2 − 7x + 12 = 0

−1, 6

0, 2

− 1 — 2 , 5

b. Graph the related function for each equation.

Identify the zeros of the function.

CRITICAL THINKING In Exercises 80–82, fi nd all values of k for which the equation has (a) two solutions, (b) one solution, and (c) no solutions.

80. 2x2 + x + 3k = 0 81. x2 − 4kx + 36 = 0

82. kx2 + 5x − 16 = 0

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencySolve the system of linear equations using any method. Explain why you chose the method. (Section 5.1, Section 5.2, and Section 5.3)

83. y = −x + 4 84. x = 16 − 4y

y = 2x − 8 3x + 4y = 8

85. 2x − y = 7 86. 3x − 2y = −20

2x + 7y = 31 x + 1.2y = 6.4

Reviewing what you learned in previous grades and lessons

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